TRIPLE FACTORIZATION OF SOME RIORDAN MATRICES

In a recent paper [2], techniques are discussed that assist in finding closed-form expressions for the formal power series for a select, but large, set of combinatorial sequences. The methods involve using infinite matrices and the Riordan group. The Riordan group is defined in section 2 of this paper. Each matrix, L, in the Riordan group is associated with a combinatorial sequence and with a matrix, SL, called the Stieltjes matrix. SL is defined in section 3. In this paper, we show that when SL is tridiagonal, then L= PCF, where the first factor P is a Pascal-type matrix, the second factor C involves the generating function for the Catalan numbers, and the third factor F involves the Fibonacci generating function. The following is an example: